Parameter estimation and identifiability, uncertainty, and sensitivity analyses for a dengue model

Implemented model
Human population
with
Mosquito population
with
Taken from
Lizarralde-Bejarano DP, Rojas-Díaz D, Arboleda-Sánchez S, Puerta-Yepes ME (2020) Sensitivity, uncertainty and identifiability analyses to define a dengue transmission model with real data of an endemic municipality of Colombia. PLOS ONE 15(3): e0229668. https://doi.org/10.1371/journal.pone.0229668
Real data
We implemented the number of dengue cases per week reported for Itagüí, a municipality in Colombia, during the outbreak that occurred in 2010.
load('Itagui.mat');
ydata = Itagui(1:40)';
D = length(ydata);
Domain = [0 D-1];
xdata = 0 : D-1;
clf
plot(ydata, '-o', 'LineWidth',1.5)
title('Number of cases of Dengue in Itagüí-Colombia')
ylabel('Number of cases')
xlabel('Time (weeks)')
grid on
Create the model as a table for GSUA-CSB toolbox implementation
First, define the model as a symbolic structure
syms Ms(t) Me(t) Mi(t) Hs(t) He(t) Hi(t) Hr(t) Hit(t)
syms lambda beta_m mu_m theta_m mu_h beta_h theta_h...
gamma_h
H = Hs + He + Hi + Hr;
M = Ms + Me + Mi;
ode1 = diff(Ms) == lambda - beta_m * Hi * Ms/H - mu_m * Ms;
ode2 = diff(Me) == beta_m * Hi * Ms/H - (theta_m + mu_m) * Me;
ode3 = diff(Mi) == theta_m * Me - mu_m * Mi;
ode4 = diff(Hs) == - beta_h * Mi * Hs/M + (He + Hi + Hr) * mu_h;
ode5 = diff(He) == beta_h * Mi * Hs/M - (theta_h + mu_h) * He;
ode6 = diff(Hi) == theta_h * He - (gamma_h + mu_h) * Hi;
ode7 = diff(Hr) == gamma_h * Hi - mu_h * Hr;
ode8 = diff(Hit) == theta_h * He;
odes = [ode1; ode2 ;ode3; ode4; ode5 ;ode6; ode7; ode8];
vars = [Hit Hi Me Hr Hs He Ms Mi];
opts = odeset('NonNegative',1:8);
Then, define the factor intervals with their corresponding names.
Range1 = [17 17 %Hit0
17 17 %Hi
0 100 %Me
0 100 %Hr
50000 250000 %Hs
0 1000 %He
0 120000 %Ms
0 100 %Mi
];
Range2 = [0 4 %beta_h
0 4 %beta_m
0.4 1.8 %gamma_h
0 42694 %Lambda
0.0004 0.0004 %mu_h
0.01 0.6 %mu_m
0.7 1.75 %theta_h
0.58 0.88 %tetha_m
];
Range = [Range1; Range2];
ParNames = {'\lambda'; '\beta_m'; '\mu_m'; '\theta_m'; '\beta_h'; '\theta_h'; '\gamma_h'};
VarNames = {'M_e(0)'; 'H_r(0)'; 'H_s(0)'; 'H_e(0)'; 'M_s(0)'; 'M_i(0)'};
FullNames = [VarNames;ParNames];
Finally, use the gsua_dpmat to create the functional table
[T,~] = gsua_dpmat(odes,vars,[0 D-1],'Model','output',2,'opt',opts,'Range',Range);
Introduce ranges in the following order:
ans(t) = 
T.Row = ["M_e(0)"; 'H_r(0)'; 'H_s(0)'; 'H_e(0)'; 'M_s(0)'; 'M_i(0)';...
'\beta_h';'\beta_m'; '\gamma_h';'\Lambda'; '\mu_m'; '\theta_h'; '\theta_m'];
Parameter estimation
First, define the optimization process options: Solver, maximum function evaluation, maximum iterations per estimation, among others.
Solver = 'fmincon';
Opt = optimoptions('fmincon','UseParallel', false,'MaxFunctionEvaluations',30000, ...
'MaxIterations', 3000, 'Display','off');
Define the number of parameter estimations to perform and define the vectors to save the estimated parameters and their respective cost value, i.e., residual.
N = 5000;
Residual = zeros(N,1);
Params = zeros(N,13);
parfor i = 1:N
[T_est, ResTemp] = gsua_pe(T,xdata, ydata,'solver',Solver,'N',1,'opt',Opt,'save',false);
Residual(i) = ResTemp;
Params(i,:) = T_est.Estfmincon;
end
Sort each set of parameters according to each residual value
[ResidualSorted, idx] = sort(Residual);
EstimatedParams = Params(idx, :);
save("Results/Estimations/Estimations_Itagui4.mat", "EstimatedParams", "ResidualSorted")
Load the estimated factors and sorted by residual value
addpath('Results')
EstimatedParamsFinal = [];
ResidualFinal = [];
for i = 1:4
load("Results/Estimations/Estimations_Itagui" + num2str(i) + ".mat");
EstimatedParamsFinal = [EstimatedParams; EstimatedParamsFinal];
ResidualFinal = [ResidualSorted; ResidualFinal];
end
[~, idx] = sort(ResidualFinal);
EstimatedParams = EstimatedParamsFinal(idx, :);
Simulate some factor estimations
We selected 4000-factor estimations to simulate. Those curves follow the trend of real data, the remaining 2000 present aberrant behavior.
N = 4000;
ysim = zeros(length(xdata), N);
for i = 1 : N
ysim(:, i) = gsua_deval(EstimatedParams(i,:), T, xdata);
end
close all
figure
plot(xdata, ysim, 'b')
hold on
plot(xdata, ydata, 'r','Linewidth', 2)
title('Factor estimtions for dengue model')
xlabel('Time (weeks)', 'FontSize', 13)
ylabel('Number of cases', 'FontSize', 13)
xlim([0 39])
set(gca,'FontSize',13)
grid on
Best factor estimation curve
Best = ysim(:, 1);
BestParams = EstimatedParams(1,:);
save("Results/Estimations/Best_Itagui.mat", "BestParams","Best")
close all
figure
hold on
plot(xdata, ydata, 'r','Linewidth', 2)
plot(xdata, Best, 'b','Linewidth', 2)
title('Best estimation for dengue model')
xlabel('Time (weeks)', 'FontSize', 13)
ylabel('Number of cases', 'FontSize', 13)
set(gca,'FontSize',13)
grid on
Identifiability analysis for 4000-factor estimations
Using the gsua_ia we obtain a table with: a nominal value of each factor as the median of the estimations, the parametric CI using the following equation:
T_ia = gsua_ia(T, EstimatedParams(1:N,:)',false,false);
T = 13×4 table
 RangeNominalEstindex
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2 H_r(0)42.951344.866943.909115.60980.001927.534245.704946.918546.918521.190529.733945.663845.663828.203257.39660.001944.335344.33530.46700.467024.875151.579049.165249.165225.939825.939818.979249.663322.322922.32297.402449.81710.001951.116063.156563.156542.469341.414141.414127.711845.357220.720018.869235.684142.501765.604955.095245.415244.751223.632845.082335.40737.71277.71270.343349.026949.026946.252744.059544.059540.59684.465846.163943.778743.778720.172641.120448.349271.611260.075860.755143.687326.26260.365848.679048.472948.472936.772711.46380.123030.192178.363048.467148.30461.0000e-031.0000e-039.9999e-049.9999e-0444.941348.991844.514430.860148.820224.866346.464593.130531.936931.936948.319846.534446.534448.802848.802847.288247.288248.356863.369463.369482.894750.161130.471324.888725.713625.713630.15670.011929.26018.8025e-0416.78892.55102.55109.02189.021845.001045.001025.373231.273939.95358.43587.936545.679238.615631.215125.292788.835811.02448.20168.201645.84631.0000e-0345.653244.36998.361469.357720.363950.127221.778918.127144.041345.418846.967345.415145.415143.004243.004248.924013.773820.886845.087042.944445.414850.219542.577064.016444.666745.671545.671545.298320.34229.9998e-0419.831039.515939.515940.99075.685324.297624.297617.762517.762542.254142.254118.072218.163439.568119.015419.015442.234641.709546.028615.08569.753319.927760.83633.870051.614651.614616.542415.722617.80481.187057.25734.944615.759515.759568.791168.791122.832776.49029.9999e-049.9999e-0462.08200.49060.490638.422738.422738.129834.473841.430812.952712.952729.517329.613614.131938.788919.045639.799712.170912.170911.150623.51520.002310.209518.20719.82263.32723.327242.611142.611116.423412.56434.029336.673329.887635.635112.812627.050512.386653.819442.295442.295421.917821.917882.516582.516512.476348.671648.671621.63874.05914.05914.16652.055411.909111.909134.019610.268348.557415.761138.435237.677942.319142.319148.396137.516846.706145.183127.767527.76759.710912.961212.961274.407751.391834.225468.889468.889414.15589.9999e-0411.956245.87329.9999e-0449.744862.934045.674445.674434.706814.503738.438434.066734.066715.538449.61900.966145.17299.9999e-0426.376176.996219.120164.54632.493748.427737.75140.042258.42678.648210.572139.080948.082748.082795.677349.064715.737334.807249.871947.658413.231698.637547.181184.118119.933519.933549.033747.834747.83479.8917e-0435.742311.569445.086745.005145.00519.9999e-049.9999e-048.080051.916911.37941.0000e-0348.622047.40281.0000e-0366.31552.98940.00757.444890.95439.9999e-0414.664014.66402.80802.808013.101747.80971.0000e-0313.688344.429238.430033.863419.94209.9961e-0469.088034.639334.639332.910732.910733.493229.99995.547248.39489.9999e-0447.320349.620949.62099.9999e-049.9999e-0442.71343.45693.456951.89530.06819.6417e-049.6417e-0445.45579.9999e-049.9999e-0486.127436.396340.701131.844548.194548.194547.942846.838548.563874.339574.339550.244249.692561.860761.86072.77682.776847.940047.057548.684551.258941.325541.325546.647447.418969.605945.100745.10071.420148.86081.0000e-030.924128.420649.454683.306269.088249.800752.731345.230848.418449.716430.298545.223422.371349.861947.751015.462846.311346.311345.440329.628649.85659.86809.868038.320650.387650.387648.035089.658445.132249.945633.916027.09108.978245.460343.95519.701846.644546.644545.324943.799529.279932.379532.379548.236745.840548.755048.055948.055947.595843.023191.997025.949645.83009.099926.576726.576748.363065.41358.438644.269028.335537.747144.935841.766340.301840.301826.70886.580340.579640.579648.234949.846920.695347.674647.128844.323476.705949.068849.068840.461948.88655.015525.715537.145147.836147.836148.131010.504528.763456.49500.00100.00102.44729.328547.406146.895549.574349.574350.461550.461530.345430.345435.881147.57846.07633.92383.923893.700537.02896.232942.035011.395649.62409.9870e-049.9870e-0434.615451.499051.49903.195948.72208.256325.562925.562945.230639.818445.190845.19085.03435.034342.62162.28502.285044.549541.501648.880148.892642.738860.256238.686420.159220.159249.21306.2105e-0410.71149.009943.893843.893814.055280.949425.716725.716780.155843.839923.189429.891532.213617.068463.755619.586346.180971.599350.344441.211632.394648.338339.14500.00160.00164.01724.01723.4553e-0546.818441.762343.014243.01420.06060.060642.622245.100942.908141.586125.988047.802750.079150.079141.022648.802946.887836.136646.60250.001035.492440.094321.377146.248048.753948.786322.068746.659217.442731.242047.397649.535543.712743.712745.525227.216947.705841.98880.53560.535625.759025.759047.181537.987793.863848.537249.087948.220348.220348.606248.606244.710244.710289.03576.802050.325412.484512.484585.83299.964548.432948.432951.407648.671542.99879.9986e-0447.91010.008740.175349.955549.955528.63946.96736.967388.582748.541441.234541.234538.897747.910646.706446.706440.508447.561948.455248.45520.005148.062948.772848.772850.664653.022153.022182.854076.837276.83725.416748.548048.041148.246849.727643.907675.983120.080320.080349.483737.396675.845450.627279.944949.488749.488747.931849.093245.15332.34312.343155.1352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5 M_s(0)7.2885e+047.5110e+047.3998e+042.3699e+041.1274e+051.0093e+059.9487e+048.8290e+048.8290e+045.1416e+046.3183e+041.0205e+051.0205e+056.6966e+048.6911e+044.1317e+041.1370e+051.1370e+059.3567e+049.3567e+048.2397e+041.0704e+051.1414e+051.1414e+056.1493e+046.1493e+044.7230e+043.3662e+048.0450e+048.0450e+043.3815e+043.6527e+044.9207e+044.3583e+043.7166e+043.7166e+049.3326e+041.1263e+051.1263e+058.7349e+044.0813e+045.4777e+041.1257e+055.2715e+049.1563e+048.3964e+049.8300e+041.1507e+054.1623e+049.8993e+041.1553e+059.2569e+041.0952e+051.0952e+055.1608e+048.8680e+048.8680e+048.6126e+041.1713e+051.1713e+059.3769e+045.9675e+041.0797e+051.1155e+051.1155e+051.9871e+045.4415e+041.1257e+059.5384e+041.0163e+056.2442e+041.0302e+051.1638e+057.2736e+044.8684e+044.7729e+044.7729e+041.0823e+057.3241e+048.8624e+044.4192e+041.1551e+056.8907e+044.4407e+047.7358e+047.7358e+046.0435e+048.0860e+047.0246e+041.0774e+055.3564e+044.8403e+041.0134e+054.6346e+047.8770e+044.2025e+043.5797e+043.5797e+048.8595e+044.3318e+044.3318e+049.4597e+049.4597e+042.2282e+042.2282e+048.0895e+049.9396e+049.9396e+041.1164e+057.1793e+047.3988e+047.0907e+044.3764e+044.3764e+045.5317e+043.2713e+047.3868e+045.9224e+048.6834e+047.0059e+047.0059e+044.8602e+044.8602e+043.7569e+043.7569e+047.1434e+048.1316e+044.7835e+049.8102e+041.1991e+058.9775e+044.5712e+046.5736e+042.0619e+049.4854e+045.6905e+046.9781e+046.9781e+041.1328e+059.8098e+044.2785e+048.3644e+041.0244e+051.0368e+053.0423e+046.5722e+049.0166e+041.0641e+057.3838e+044.8427e+042.4058e+049.5002e+039.5002e+035.0925e+045.0925e+047.7439e+041.1159e+051.0996e+053.7136e+046.6534e+042.9724e+047.4699e+041.7889e+049.9899e+047.1322e+046.5994e+046.5994e+043.6699e+049.1863e+048.5630e+041.0010e+059.6647e+049.6647e+041.0325e+056.0228e+047.5323e+047.5323e+049.0876e+049.0876e+041.0166e+051.0166e+051.0600e+051.0709e+051.1571e+058.4974e+048.4974e+041.2264e+047.0909e+041.0282e+052.9782e+042.3313e+046.7321e+047.4983e+046.7180e+044.5498e+044.5498e+049.0391e+043.4525e+049.7659e+042.7513e+045.7784e+041.0568e+058.5704e+048.5704e+048.6316e+048.6316e+041.1527e+051.1125e+057.7650e+047.7650e+047.5552e+041.0721e+051.0721e+056.7780e+046.7780e+043.2913e+042.6541e+041.1182e+056.8734e+046.8734e+049.3866e+047.8124e+042.3561e+049.6392e+043.8904e+044.9081e+041.1373e+051.1373e+056.2485e+041.1225e+051.0816e+054.6251e+043.0717e+049.1738e+047.4398e+047.4398e+045.2560e+045.2560e+044.3154e+048.8671e+046.0875e+045.2395e+045.1630e+042.6940e+043.7614e+046.5378e+041.0311e+058.5304e+041.0003e+051.0003e+058.7651e+048.7651e+044.9066e+044.9066e+041.0421e+058.6609e+048.6609e+043.7357e+047.2185e+047.2185e+047.1149e+049.9476e+049.8513e+049.8513e+045.4631e+047.7226e+048.8131e+041.0446e+055.8499e+043.2390e+048.7583e+048.7583e+047.4342e+041.0941e+053.7118e+046.3726e+043.0428e+043.0428e+047.6353e+049.7996e+049.7996e+044.2118e+049.7294e+042.2986e+047.3006e+047.3006e+048.5604e+049.4964e+041.0616e+054.4170e+045.6618e+046.1935e+045.5117e+041.1147e+051.1147e+059.9910e+046.3357e+041.1156e+055.3686e+045.3686e+044.6353e+045.5344e+041.0659e+051.9818e+043.4014e+046.9647e+046.0919e+041.1832e+054.4309e+045.2163e+044.9217e+046.6943e+042.4415e+048.7985e+045.6216e+046.8975e+049.5269e+045.9044e+045.9044e+041.1273e+058.7533e+043.2448e+041.0940e+055.8243e+044.8961e+041.0908e+058.0934e+048.0952e+044.4348e+047.7661e+047.7661e+045.1611e+046.2690e+046.2690e+049.8257e+046.9101e+041.0964e+053.7142e+049.9476e+049.9476e+044.9788e+044.9788e+041.0205e+058.5305e+045.3519e+041.9993e+043.4752e+047.3095e+049.0103e+049.3388e+046.2672e+046.5165e+041.0881e+051.1336e+059.8356e+041.5330e+041.5330e+049.0379e+049.0379e+046.2524e+042.6413e+049.4417e+048.4358e+041.0523e+056.2341e+048.4985e+048.3810e+046.3102e+047.3405e+045.4601e+045.4601e+041.1644e+051.1644e+056.6551e+045.6044e+049.4054e+047.4645e+041.0854e+058.8296e+047.9550e+047.9550e+041.1401e+051.1401e+059.2847e+049.3330e+049.3330e+048.7170e+048.1865e+043.9581e+043.9581e+044.4140e+048.2292e+048.2292e+047.5239e+045.8106e+042.0781e+044.7848e+045.7346e+045.7346e+048.5325e+042.3677e+041.1589e+052.4391e+042.4391e+046.2354e+047.0194e+042.7467e+042.7467e+041.1492e+051.1492e+052.1774e+041.0081e+053.2067e+048.5270e+043.4799e+043.4799e+047.1662e+048.7065e+048.8213e+048.6993e+048.6993e+041.1130e+055.1331e+048.8639e+044.5972e+045.9322e+046.8658e+041.1300e+051.1853e+059.4583e+048.5351e+048.7026e+049.8306e+043.3730e+042.2303e+045.1279e+044.9378e+048.8652e+045.6900e+041.0903e+051.0895e+051.0895e+053.8755e+041.0426e+057.8684e+049.0641e+049.0641e+042.7884e+048.0582e+048.0582e+048.3665e+048.0303e+049.5586e+047.7593e+041.0771e+058.9764e+048.6151e+047.4780e+048.4186e+049.0681e+047.3921e+047.3921e+046.0284e+048.8093e+046.2397e+048.4327e+048.4327e+047.0068e+042.7927e+041.0348e+059.8072e+049.8072e+046.8863e+041.0237e+057.9080e+045.3662e+045.1604e+045.1912e+045.2731e+045.2731e+041.1329e+058.7467e+041.1296e+056.0287e+044.5710e+046.0418e+049.0939e+047.7386e+047.7366e+047.7366e+044.5912e+044.4778e+047.6726e+047.6726e+047.5610e+049.8278e+046.0938e+047.8076e+049.3867e+043.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Uncertainty analysis for parametric CI
clf
N = 1000;
M = gsua_dmatrix(T_ia,N,'Method','Sobol');
gsua_ua(M, T_ia,'xdata', xdata, 'ynom', ydata, 'parallel', false);
Progress: 100% Estimated processing time (h:m:s): 0:0:9 Remaining time (h:m:s): 0:0:0 Elapsed time (h:m:s): 0:0:9 Estimated stop time (h:m:s): 17:42:50 Number of simulations: 2000
Sensitivity analysis (SA): estimation intervals
Perform the SA using the estimation intervals, i.e., the estimation box
N = 10000;
M = gsua_dmatrix(T,N,'Method','Sobol');
T_sa = gsua_sa(M, T);
Starting parallel pool (parpool) using the 'local' profile ... Connected to the parallel pool (number of workers: 8). Progress: 7% Estimated processing time (h:m:s): 0:7:38 Remaining time (h:m:s): 0:7:7 Elapsed time (h:m:s): 0:0:30 Estimated stop time (h:m:s): 17:55:20 Number of simulations: 75000 Progress: 14% Estimated processing time (h:m:s): 0:4:56 Remaining time (h:m:s): 0:4:17 Elapsed time (h:m:s): 0:0:39 Estimated stop time (h:m:s): 17:52:38 Number of simulations: 75000 Progress: 20% Estimated processing time (h:m:s): 0:3:44 Remaining time (h:m:s): 0:2:59 Elapsed time (h:m:s): 0:0:44 Estimated stop time (h:m:s): 17:51:26 Number of simulations: 75000 Progress: 27% Estimated processing time (h:m:s): 0:3:7 Remaining time (h:m:s): 0:2:17 Elapsed time (h:m:s): 0:0:50 Estimated stop time (h:m:s): 17:50:49 Number of simulations: 75000 Progress: 34% Estimated processing time (h:m:s): 0:2:46 Remaining time (h:m:s): 0:1:51 Elapsed time (h:m:s): 0:0:55 Estimated stop time (h:m:s): 17:50:28 Number of simulations: 75000 Progress: 40% Estimated processing time (h:m:s): 0:2:34 Remaining time (h:m:s): 0:1:32 Elapsed time (h:m:s): 0:1:1 Estimated stop time (h:m:s): 17:50:16 Number of simulations: 75000 Progress: 47% Estimated processing time (h:m:s): 0:2:25 Remaining time (h:m:s): 0:1:17 Elapsed time (h:m:s): 0:1:7 Estimated stop time (h:m:s): 17:50:7 Number of simulations: 75000 Progress: 54% Estimated processing time (h:m:s): 0:2:17 Remaining time (h:m:s): 0:1:4 Elapsed time (h:m:s): 0:1:13 Estimated stop time (h:m:s): 17:49:59 Number of simulations: 75000 Progress: 60% Estimated processing time (h:m:s): 0:2:10 Remaining time (h:m:s): 0:0:52 Elapsed time (h:m:s): 0:1:18 Estimated stop time (h:m:s): 17:49:52 Number of simulations: 75000 Progress: 67% Estimated processing time (h:m:s): 0:2:6 Remaining time (h:m:s): 0:0:42 Elapsed time (h:m:s): 0:1:24 Estimated stop time (h:m:s): 17:49:48 Number of simulations: 75000 Progress: 74% Estimated processing time (h:m:s): 0:2:2 Remaining time (h:m:s): 0:0:32 Elapsed time (h:m:s): 0:1:29 Estimated stop time (h:m:s): 17:49:44 Number of simulations: 75000 Progress: 81% Estimated processing time (h:m:s): 0:1:59 Remaining time (h:m:s): 0:0:23 Elapsed time (h:m:s): 0:1:35 Estimated stop time (h:m:s): 17:49:41 Number of simulations: 75000 Progress: 87% Estimated processing time (h:m:s): 0:1:56 Remaining time (h:m:s): 0:0:15 Elapsed time (h:m:s): 0:1:40 Estimated stop time (h:m:s): 17:49:38 Number of simulations: 75000 Progress: 94% Estimated processing time (h:m:s): 0:1:54 Remaining time (h:m:s): 0:0:7 Elapsed time (h:m:s): 0:1:46 Estimated stop time (h:m:s): 17:49:35 Number of simulations: 75000 Progress: 100% Estimated processing time (h:m:s): 0:1:52 Remaining time (h:m:s): 0:0:0 Elapsed time (h:m:s): 0:1:52 Estimated stop time (h:m:s): 17:49:33 Number of simulations: 75000
sum(T_sa.Si)/sum(abs(T_sa.Si))
ans = 1.0000
clf
gsua_plot('Pie',T_sa,T_sa.STi)
SA: parametric confident intervals
Perform the SA using the confident intervals using the median value of the parameters
N = 10000;
M = gsua_dmatrix(T_ia,N,'Method','Sobol');
T_sa = gsua_sa(M, T_ia);
Progress: 7% Estimated processing time (h:m:s): 0:1:22 Remaining time (h:m:s): 0:1:16 Elapsed time (h:m:s): 0:0:5 Estimated stop time (h:m:s): 17:57:5 Number of simulations: 75000 Progress: 14% Estimated processing time (h:m:s): 0:1:21 Remaining time (h:m:s): 0:1:10 Elapsed time (h:m:s): 0:0:10 Estimated stop time (h:m:s): 17:57:4 Number of simulations: 75000 Progress: 20% Estimated processing time (h:m:s): 0:1:22 Remaining time (h:m:s): 0:1:5 Elapsed time (h:m:s): 0:0:16 Estimated stop time (h:m:s): 17:57:5 Number of simulations: 75000 Progress: 27% Estimated processing time (h:m:s): 0:1:22 Remaining time (h:m:s): 0:1:0 Elapsed time (h:m:s): 0:0:22 Estimated stop time (h:m:s): 17:57:5 Number of simulations: 75000 Progress: 34% Estimated processing time (h:m:s): 0:1:23 Remaining time (h:m:s): 0:0:55 Elapsed time (h:m:s): 0:0:27 Estimated stop time (h:m:s): 17:57:6 Number of simulations: 75000 Progress: 40% Estimated processing time (h:m:s): 0:1:24 Remaining time (h:m:s): 0:0:50 Elapsed time (h:m:s): 0:0:33 Estimated stop time (h:m:s): 17:57:7 Number of simulations: 75000 Progress: 47% Estimated processing time (h:m:s): 0:1:24 Remaining time (h:m:s): 0:0:45 Elapsed time (h:m:s): 0:0:39 Estimated stop time (h:m:s): 17:57:7 Number of simulations: 75000 Progress: 54% Estimated processing time (h:m:s): 0:1:25 Remaining time (h:m:s): 0:0:40 Elapsed time (h:m:s): 0:0:45 Estimated stop time (h:m:s): 17:57:9 Number of simulations: 75000 Progress: 60% Estimated processing time (h:m:s): 0:1:27 Remaining time (h:m:s): 0:0:34 Elapsed time (h:m:s): 0:0:52 Estimated stop time (h:m:s): 17:57:10 Number of simulations: 75000 Progress: 67% Estimated processing time (h:m:s): 0:1:27 Remaining time (h:m:s): 0:0:29 Elapsed time (h:m:s): 0:0:58 Estimated stop time (h:m:s): 17:57:10 Number of simulations: 75000 Progress: 74% Estimated processing time (h:m:s): 0:1:27 Remaining time (h:m:s): 0:0:23 Elapsed time (h:m:s): 0:1:4 Estimated stop time (h:m:s): 17:57:10 Number of simulations: 75000 Progress: 81% Estimated processing time (h:m:s): 0:1:27 Remaining time (h:m:s): 0:0:17 Elapsed time (h:m:s): 0:1:10 Estimated stop time (h:m:s): 17:57:10 Number of simulations: 75000 Progress: 87% Estimated processing time (h:m:s): 0:1:27 Remaining time (h:m:s): 0:0:11 Elapsed time (h:m:s): 0:1:15 Estimated stop time (h:m:s): 17:57:10 Number of simulations: 75000 Progress: 94% Estimated processing time (h:m:s): 0:1:27 Remaining time (h:m:s): 0:0:5 Elapsed time (h:m:s): 0:1:21 Estimated stop time (h:m:s): 17:57:10 Number of simulations: 75000 Progress: 100% Estimated processing time (h:m:s): 0:1:27 Remaining time (h:m:s): 0:0:0 Elapsed time (h:m:s): 0:1:27 Estimated stop time (h:m:s): 17:57:10 Number of simulations: 75000
sum(T_sa.Si)/sum(abs(T_sa.Si))
ans = 0.9994
clf
gsua_plot('Pie',T_sa,T_sa.STi)